An important question for proponents of non-contractive approaches to paradox is why contraction fails. Zardini offers an answer, namely that paradoxical sentences exhibit a kind of instability. I elaborate this idea using revision theory, and I argue that while instability does motivate failures of contraction, it equally motivates failure of many principles that non-contractive theorists want to maintain.

Self-referential sentences have troubled our understanding of language for centuries. The most famous self-referential sentence is probably the Liar, a sentence that says of itself that it is false. The Liar Paradox has encouraged many philosophers to establish theories of truth that manage to give a proper account of the truth predicate in a formal language. Kripke’s Fixed Point Theory from 1975 is one famous example of such a formal theory of truth that aims at giving a plausible notion of (...) truth by allowing truth value gaps. However, not only the concept of truth gives rise to paradoxes. A syntactical treatment of epistemic notions like belief and knowledge leads to contradictions that very much resemble the Liar Paradox. Therefore, it seems to be fruitful to apply the established theories of truth to epistemic concepts. In this paper, I will present one such attempt of solving the epistemic paradoxes: I adapt Kripke’s Fixed Point Theory and interpret truth, knowledge and belief within the framework of a partial logic. Thereby I find not only the fixed point of truth but also the fixed points of knowledge and belief. In this fixed point, the predicates of truth, belief and knowledge find their definite interpretation and the paradoxes are avoided. (shrink)

The sentences employed in semantic paradoxes display a wide range of semantic behaviours. However, the main theories of truth currently available either fail to provide a theory of paradox altogether, or can only account for some paradoxical phenomena by resorting to multiple interpretations of the language. In this paper, I explore the wide range of semantic behaviours displayed by paradoxical sentences, and I develop a unified theory of truth and paradox, that is a theory of truth that also provides a (...) unified account of paradoxical sentences. The theory I propose here yields a threefold classification of paradoxical sentences – liar-like sentences, truth-teller-like sentences, and revenge sentences. Unlike existing treatments of semantic paradox, the theory put forward in this paper yields a way of interpreting all three kinds of paradoxical sentences, as well as unparadoxical sentences, within a single model. (shrink)

The model of self-referential truth presented in this paper, named Revision-theoretic supervaluation, aims to incorporate the philosophical insights of Gupta and Belnap’s Revision Theory of Truth into the formal framework of Kripkean fixed-point semantics. In Kripke-style theories the final set of grounded true sentences can be reached from below along a strictly increasing sequence of sets of grounded true sentences: in this sense, each stage of the construction can be viewed as an improvement on the previous ones. I want to (...) do something similar replacing the Kripkean sets of grounded true sentences with revision-theoretic sets of stable true sentences. This can be done by defining a monotone operator through a variant of van Fraassen’s supervaluation scheme which is simply based on ω-length iterations of the Tarskian operator. Clearly, all virtues of Kripke-style theories are preserved, and we can also prove that the resulting set of “grounded” true sentences shares some nice features with the sets of stable true sentences which are provided by the usual ways of formalising revision. What is expected is that a clearer philosophical content could be associated to this way of doing revision; hopefully, a content directly linked with the insights underlying finite revision processes. (shrink)

The classical theory of definitions bans so-called circular definitions, namely, definitions of a unary predicate P, based on stipulations of the form $$Px =_{\mathsf {Df}} \phi,$$where ϕ is a formula of a fixed first-order language and the definiendumP occurs into the definiensϕ. In their seminal book The Revision Theory of Truth, Gupta and Belnap claim that “General theories of definitions are possible within which circular definitions [...] make logical and semantic sense” [p. IX]. In order to sustain their claim, they (...) develop in this book one general theory of definitions based on revision sequences, namely, ordinal-length iterations of the operator which is induced by the definition of the predicate. Gupta-Belnap’s approach to circular definitions has been criticised, among others, by D. Martin and V. McGee. Their criticisms point to the logical complexity of revision sequences, to their relations with ordinary mathematical practice, and to their merits relative to alternative approaches. I will present an alternative general theory of definitions, based on a combination of supervaluation and ω-length revision, which aims to address some criticisms raised against revision sequences, while preserving the philosophical and mathematical core of revision. (shrink)

We study, in an abstract and general framework, formal representations of dependence and groundedness which occur in semantic theories of truth. Our goals are: (a) to relate the different ways in which groundedness is defined according to the way dependence is represented; and (b) to represent different notions of dependence as instances of a suitable generalisation of the mathematical notion of functional dependence.

Kremer presented three approaches of comparing fixed-point and revision theories of truth in Kremer, 363–403, 2009). Using these approaches, he established the relationships among ten fixed-point theories suggested by Kripke in, 690–716, 1975) and three revision theories presented by Gupta and Belnap in. This paper continues Kremer’s work. We add five other revision theories to the comparisons, including the theory proposed by Gupta in, 1–60, 1982), the theory proposed by Herzberger in, 61–102, 1982), the theory based on fully-varied revision sequences (...) proposed by Gupta and Belnap in, the theory proposed by Yaqūb in, and the theory based on weakly consistent revision sequences. We show that, the notion of Thomason model defined by Belnap’s limit rule is not equivalent to the one defined by Gupta’s limit rule, and that the theory based on fully-varied revision sequences is ≤2-equivalent to the one based on the greatest intrinsic fixed point of σ. (shrink)

In The Revision Theory of Truth (MIT Press), Gupta and Belnap (1993) claim as an advantage of their approach to truth "its consequence that truth behaves like an ordinary classical concept under certain conditions—conditions that can roughly be characterized as those in which there is no vicious reference in the language." To clarify this remark, they define Thomason models, nonpathological models in which truth behaves like a classical concept, and investigate conditions under which a model is Thomason: they argue that (...) a model is Thomason when there is no vicious reference in it. We extend their investigation, considering notions of nonpathologicality and senses of "no vicious reference" generated both by revision theories of truth and by fixedpoint theories of truth. We show that some of the fixed-point theories have an advantage analogous to that which Gupta and Belnap claim for their approach, and that at least one revision theory does not. This calls into question the claim that the revision theories have a distinctive advantage in this regard. (shrink)

In various theories of truth, people have set forth many definitions to clarify in what sense a set of sentences is paradoxical. But what, exactly, is _a_ paradox per se? It has not yet been realized that there is a gap between ‘being paradoxical’ and ‘being a paradox’. This paper proposes that a paradox is a minimally paradoxical set meeting some closure property. Along this line of thought, we give five tentative definitions based upon the folk notion of paradoxicality implied (...) in Tarski’s undefinability theorem of truth and the dependence relation proposed by Leitgeb (J Philos Log 34(2):155–192, 2005). A definition of ‘being a paradox’ is acceptable if its extension must be sufficiently comprehensive to include all known paradoxes on the one hand, and its standard must also be high enough to exclude those that are evidently not paradoxes on the other hand. It turns out that only the last attempt is acceptable: a set of sentences is a paradox if it is paradoxical and self-dependent, and at the same time, it and any of its paradoxical and self-dependent subsets bisimulate each other. In this way, we articulate in what sense a set of sentences is a paradox, and so establish a connection between the notion of a paradox and the notion of paradoxicality. (shrink)

Semantic and soritical paradoxes display remarkable family resemblances. For one thing, several non-classical logics have been independently applied to both kinds of paradoxes. For another, revenge paradoxes and higher-order vagueness—among the most serious problems targeting solutions to semantic and soritical paradoxes—exhibit a rather similar dynamics. Some authors have taken these facts to suggest that truth and vagueness require a unified logical framework, or perhaps that the truth predicate is itself vague. However, a common core of semantic and soritical paradoxes has (...) not been identified yet, and no explanation of their relationships has been provided. Here we aim at filling this lacuna, in the framework of many-valued logics. We provide a unified diagnosis of semantic and soritical paradoxes, identifying their source in a general form of indiscernibility. We then develop our diagnosis into a theory of paradoxicality, which formalizes both semantic and soritical paradoxes as arguments involving specific instances of our generalized indiscernibility principle, and correctly predicts which logics can non-trivially solve them. (shrink)

§1. Introduction. When truth-theoretic paradoxes are generated, two factors seem to be at play: the behaviour that truth intuitively has; and the facts about which singular terms refer to which sentences, and so on. For example, paradoxicality might be partially attributed to the contingent fact that the singular term, "the italicized sentence on page one", refers to the sentence, The italicized sentence on page one is not true. Factors of this second kind might be represented by a ground model: an (...) interpretation of all the names, function symbols, and predicates in the potentially self-referential language under study, with the exception of the predicate "x is true". Formally, suppose that L is an uninterpreted first order language. M = 〈D, I〉 is a classical model for L if D is a nonempty set; and I is a function assigning to each name of L a member of D, to each n-place function symbol of L a function from Dn to D, and to each nonlogical n-place predicate of L a function from Dn to {t, f}. Suppose furthermore that L+ is obtained by adding a new one-place predicate T to L, and that L has a quote name ‘A’ for each sentence A of L+. We follow Gupta and Belnap [5] in defining S =df {A: A is a sentences of L+}. A classical model M = 〈D, I〉 for L is a ground model for L iff both S ⊆ D, and I(‘A’) = A for each A ∈ S. A classical ground model for L is a representation of the supposedly unproblematic fragment of L+: a representation of which terms refer to which objects, and of which objects have which nonsemantic properties. (shrink)

Philip Kremer, Department of Philosophy, McMaster University Note: The following version of this paper does not contain the proofs of the stated theorems. A longer version, complete with proofs, is forthcoming. §1. Introduction. In "The truth is never simple" and its addendum, Burgess conducts a breathtakingly comprehensive survey of the complexity of the set of truths which arise when you add a truth predicate to arithmetic, and interpret that predicate according to the fixed point semantics or the revision-theoretic semantics for (...) languages expressing their own truth concepts. Burgess considers various sets that can be said to represent truth in this context, and shows that their complexity ranges from Π11 or Σ11 to Π12 or Σ12. Thus, enriching arithmetic with a truth predicate increases its complexity, which is otherwise only ∆11. (shrink)